Physics 42200 Waves & Oscillations Lecture 21 Review Spring 2013 Semester Matthew Jones
Midterm Exam: Date: Wednesday, March 6 th Time: 8:00 10:00 pm Room: PHYS 203 Material: French, chapters 1-8
Review 1. Simple harmonic motion (one degree of freedom) mass/spring, pendulum, water in pipes, RLC circuits damped harmonic motion 2. Forced harmonic oscillators amplitude/phase of steady state oscillations transient phenomena 3. Coupled harmonic oscillators masses/springs, coupled pendula, RLC circuits 4. Uniformly distributed discrete systems masses on string fixed at both ends lots of masses/springs
Review 5. Continuously distributed systems (standing waves) string fixed at both ends sound waves in pipes (open end/closed end) transmission lines Fourier analysis 6. Progressive waves in continuous systems dispersion, phase velocity/group velocity reflection/transmission coefficients 7. Waves in two and three dimensions Laplacian operator Rotationally symmetric solutions in 2d and 3d
Simple Harmonic Motion Any system in which the force is opposite the displacement will oscillate about a point of stable equilibrium If the force is proportional to the displacement it will undergo simple harmonic motion Examples: Mass/massless spring Elastic rod (characterized by Young s modulus) Floating objects Torsion pendulum (shear modulus) Simple pendulum Physical pendulum LC circuit
Simple Harmonic Motion You should be able to draw a free-body diagram and express the force in terms of the displacement. Use Newton s law: = Write it in standard form: + = 0 Solutions are of the form: = cos = cos+sin You must be able to use the initial conditions to solve for the constants of integration
Examples sin l = /l =
Examples =?
Damped Harmonic Motion Damping forces remove energy from the system We will only consider cases where the force is proportional to the velocity: = You should be able to construct a free-body diagram and write the resulting equation of motion: + + = 0 You should be able to write it in the standard form: + +! = 0 You must be able to solve this differential equation!
Damped Harmonic Motion + +! = 0 Let = " #$ Characteristic polynomial: % + %+! = 0 Roots (use the quadratic formula): % = 2 ± Classification of solutions: 4! Over-damped: 4! > 0 (distinct real roots) Critically damped: 4 =! (one root) Under-damped: 4! < 0 (complex roots)
Damped Harmonic Motion Over-damped motion: 4! > 0 = ",-. $ " $ -. /, 0 1. +",-. $ -.,$ " /, 0 1. Under-damped motion: 4! < 0 = ",2 $ " 3$ 0 1., 2. 4 +",2 $ ",3$ 0 1., 2. 4 Critically damped motion: = (+)",-. $ You must be able to use the initial conditions to solve for the constants of integration
Example? @(A) B Sum of potential differences: 7 89 $ 8 9 : 1 < =!+> 9 8 Initial charge, =!, defines the initial conditions. C! = 0
Example 7 89 $ 8 +9 :+1 < =!+> 9 8! = 0 Differentiate once with respect to time: 7 8 9 8 +:89 8 + 1 < 9 = 0 8 9 8 + 89 8 +! 9 = 0 Remember, the solution is @(A)but the initial conditions might be in terms of D A = D E +F@ A GA (See examples from Lecture 7)
Forced Harmonic Motion Now the differential equation is + + = =! cos Driving function is not always given in terms of a force remember Question #2 on Assignment #3: H + H +! H = 8 I 8 = < cos General properties: Steady state properties: 1/ Solution is H = cos Amplitude,, and phase,, depend on
Forced Harmonic Motion Q quantifies the amount of damping: = =! (large Q means small damping force) =!!! = tan,k 1/=!!! / + 1 K/ =
Resonance Qualitative features: amplitude (! ) Q=5 Q=4 Q=3 Q=2 Q=1 NOPP! = 1 NOPP =! 4 /!
QR()!! /2 = = 10 Average Power The rate at which the oscillator absorbs energy is: QR() =!! 2= 1!! + 1 = = = 5 = = 3 Full-Width-at-Half-Max: UVWX = Y E D = Z = = 1 /!
Resonance Qualitative features: phase shift = tan,k 1/=!! 0 at low frequencies \ at high frequencies ] = ^ _ when Y = Y E
Coupled Oscillators l l Restoring force on pendulum A: ` = (` a ) Restoring force on pendulum B: a = (` a ) a ` ` + l ` + ` a = 0 a + l a ` a = 0
Coupled Oscillators You must be able to draw the free-body diagram and set up the system of equations. ` + l ` + ` a = 0 a + l a ` a = 0 You must be able to write this system as a matrix equation. ` a +! + b b b! + b `() a () = 0
Coupled Oscillators Assume solutions are of the form `() a () = ` cos a Then,! + b b ` b! + b = 0 a You must be able to calculate the eigenvalues of a 2x2 or 3x3 matrix. Calculate the determinant Calculate the roots by factoring the determinant or using the quadratic formula. These are the frequencies of the normal modes of oscillation.
Coupled Oscillators You must be able to calculate the eigenvectors of a 2x2 or 3x3 matrix General solution: c = dc K cos K e +fc cos g + You must be able to solve for the constants of integration using the initial conditions.
Example K
Example K
Example sin K K l K = K l + K l = K l sin K l K sin K l
Equations of motion: Example K = K l + K l = K l Write as a matrix: K +2! K! = 0 +!! K = 0 K + 2!!!! K = 0
Normal modes: Eigenvalue problem: +2! Example c 3 = c 3 cos! K = 0! +! +2!!! + = 0! +2! ω +! 4! = 0 j 2! j! 4! = 0 j = = 3 2! ± 1 2! 4 +4! 4
Example Eigenvector for ω K =! ( k, l )! Normal mode: 1+ 5 2 1 K = 1 1+ 5 2 = 1+ 5 2 1 1+ 5 2 K K = 0 cos K %
Example Eigenvector for ω K =! ( km l )! Normal mode: 1 5 2 1 = 1 1 5 2 = 1 5 2 1 1 5 2 K K = 0 cos n